{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:IQH47UYVPGI4M34PIAG4L2FLCH","short_pith_number":"pith:IQH47UYV","schema_version":"1.0","canonical_sha256":"440fcfd3157991c66f8f400dc5e8ab11fe1b000fd59bce64fa3ef7ff028693a9","source":{"kind":"arxiv","id":"0911.2415","version":16},"attestation_state":"computed","paper":{"title":"On congruences related to central binomial coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2009-11-12T19:15:58Z","abstract_excerpt":"It is known that $\\sum_{k=0}^\\infty\\binom{2k}{k}/((2k+1)4^k)=\\pi/2$ and $\\sum_{k=0}^\\infty\\binom{2k}{k}/((2k+1)16^k)=\\pi/3$. In this paper we obtain their p-adic analogues such as $$\\sum_{p/2<k<p}\\binom{2k}{k}/((2k+1)4^k)=3\\sum_{p/2<k<p}\\binom{2k}{k}/((2k+1)16^k)= pE_{p-3} (mod p^2),$$ where p>3 is a prime and E_0,E_1,E_2,... are Euler numbers. Besides these, we also deduce some other congruences related to central binomial coefficients. In addition, we pose some conjectures one of which states that for any odd prime p we have $$\\sum_{k=0}^{p-1}\\binom{2k}{k}^3=4x^2-2p (mod p^2)$$ if (p/7)=1 an"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0911.2415","kind":"arxiv","version":16},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-11-12T19:15:58Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"8e3778115d41665cc2c44bf0d353329615bb7a6ef8fdff376f20d5dbed5b4f74","abstract_canon_sha256":"250e99b5e142bfc9f4add04c388a230d48913229a02f4d4a40a13312102eaa60"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:16:30.944594Z","signature_b64":"YAI3SaCw+oIYj0uEE3wPUQLU4UD+W3OyssylsTP4EwRf5IbP6nvEmPYeAah9+YjFpteshONiKO5Ulyv6V/UoBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"440fcfd3157991c66f8f400dc5e8ab11fe1b000fd59bce64fa3ef7ff028693a9","last_reissued_at":"2026-05-18T04:16:30.944208Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:16:30.944208Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On congruences related to central binomial coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2009-11-12T19:15:58Z","abstract_excerpt":"It is known that $\\sum_{k=0}^\\infty\\binom{2k}{k}/((2k+1)4^k)=\\pi/2$ and $\\sum_{k=0}^\\infty\\binom{2k}{k}/((2k+1)16^k)=\\pi/3$. In this paper we obtain their p-adic analogues such as $$\\sum_{p/2<k<p}\\binom{2k}{k}/((2k+1)4^k)=3\\sum_{p/2<k<p}\\binom{2k}{k}/((2k+1)16^k)= pE_{p-3} (mod p^2),$$ where p>3 is a prime and E_0,E_1,E_2,... are Euler numbers. Besides these, we also deduce some other congruences related to central binomial coefficients. In addition, we pose some conjectures one of which states that for any odd prime p we have $$\\sum_{k=0}^{p-1}\\binom{2k}{k}^3=4x^2-2p (mod p^2)$$ if (p/7)=1 an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.2415","kind":"arxiv","version":16},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"0911.2415","created_at":"2026-05-18T04:16:30.944270+00:00"},{"alias_kind":"arxiv_version","alias_value":"0911.2415v16","created_at":"2026-05-18T04:16:30.944270+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0911.2415","created_at":"2026-05-18T04:16:30.944270+00:00"},{"alias_kind":"pith_short_12","alias_value":"IQH47UYVPGI4","created_at":"2026-05-18T12:26:00.592388+00:00"},{"alias_kind":"pith_short_16","alias_value":"IQH47UYVPGI4M34P","created_at":"2026-05-18T12:26:00.592388+00:00"},{"alias_kind":"pith_short_8","alias_value":"IQH47UYV","created_at":"2026-05-18T12:26:00.592388+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/IQH47UYVPGI4M34PIAG4L2FLCH","json":"https://pith.science/pith/IQH47UYVPGI4M34PIAG4L2FLCH.json","graph_json":"https://pith.science/api/pith-number/IQH47UYVPGI4M34PIAG4L2FLCH/graph.json","events_json":"https://pith.science/api/pith-number/IQH47UYVPGI4M34PIAG4L2FLCH/events.json","paper":"https://pith.science/paper/IQH47UYV"},"agent_actions":{"view_html":"https://pith.science/pith/IQH47UYVPGI4M34PIAG4L2FLCH","download_json":"https://pith.science/pith/IQH47UYVPGI4M34PIAG4L2FLCH.json","view_paper":"https://pith.science/paper/IQH47UYV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0911.2415&json=true","fetch_graph":"https://pith.science/api/pith-number/IQH47UYVPGI4M34PIAG4L2FLCH/graph.json","fetch_events":"https://pith.science/api/pith-number/IQH47UYVPGI4M34PIAG4L2FLCH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IQH47UYVPGI4M34PIAG4L2FLCH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IQH47UYVPGI4M34PIAG4L2FLCH/action/storage_attestation","attest_author":"https://pith.science/pith/IQH47UYVPGI4M34PIAG4L2FLCH/action/author_attestation","sign_citation":"https://pith.science/pith/IQH47UYVPGI4M34PIAG4L2FLCH/action/citation_signature","submit_replication":"https://pith.science/pith/IQH47UYVPGI4M34PIAG4L2FLCH/action/replication_record"}},"created_at":"2026-05-18T04:16:30.944270+00:00","updated_at":"2026-05-18T04:16:30.944270+00:00"}